Boats and Streams

Important facts:  

1)In water, the direction along the stream is called down stream.

2)Direction against the stream is called upstream.

3)The speed of boat in still water is U km/hr and the speed of 
stream is V km/hr then 

speed down stream =U + V km/hr
speed up stream = U – V km/hr

Formulae:

If the speed down stream is A km/hr and the speed up stream is 
B km/hr then speed in still water = ½(A+B) km/hr
rate of stream =1/2(A-B) km/hr

Problems:

1. In one hour a boat goes 11 km long the stream and 5 km 
against the stream. The speed of the boat in still water is?

Sol:
       Speed in still water = ½ ( 11+5) km/hr= 8 kmph

2.A man can row 18 kmph in still water. It takes him thrice 
as long as row up as to row down the river. find the rate 
of stream.

Sol:
       Let man's rate up stream be xkmph
    then, in still water =1/2[3x+x]=2x kmph
             so, 2x= 18, x=9
          rate upstream =9kmph
         rate downstream =27 kmph
         rate of stream = ½ [27-9]
                        = 9kmph

3.A man can row 71/2kmph in still watre . if in a river 
running at 1.5 km an hour, if takes him 50 min to row to 
place and back. how far off is the place? 

Sol:        speed down stream =7.5+1.5=9kmph
               speed upstream =7.5-1.5=6kmph
          let the required distence x km. then ,
          x/9+x/6=50/60 = 2x+3x= 5/6*18
                 5x=15,     x=3
         Hence, the required distence is 3 km

4.A man can row 3 quarters of a km aganist the stream is
111/4 min. the speed of the man in still water is ?

Sol: rate upstream = 750/625 m/sec =10/9 m/sec
       rate downstream =750/450 m/sec = 5/3 m/sec
     rate in still water =1/2[10/9+5/3] = 25/18 m/sec
                        = 25/18*18/5=5 kmph

5.A boat can travel with a speed of 13 kmph in still water.
if the speed of stream is 4 kmph,find the time taken by
the boat to go 68 km downstream?

Sol: Speed down stream = 13+4= 17 kmph
    time taken to travel 68km downstream =68/17 hrs
                              = 4 hrs
                                              

6.A boat takes 90 min less to travel 36 miles downstream then 
to travel the same distence upstream. if the speed of the 
boat in still water is 10 mph . The speed of the stream is :

Sol: Let the speed of the stream be x mph .
      then, speed downstream = [10+x]mph
          speed upstream =[10-x] mph
     36/[10+x] - 36/[10-x] = 90/60 =72x*60= 90[100-x2]
             (x+50)(x-2) =0
              x=2 kmph
7.At his usual rowing rate, Rahul 12 miles down stream in a 
certain river in 6 hrs less than it takes him to travel the
same distence upstream. but if he could double his usual 
rowing rate for his 24 miles roundthe down stream 12 miles 
would then take only one hour less than the up stream 12 miles.
what is the speed of the current in miles per hours?

Sol: Let the speed in still water be x mph and the speed of 
    the curren be y mph.
         then, speed upstream = (x-y)
         speed downstream =(x+y)
         12/(x-y) - 12/(x+y) = 6
     6(x2 – y2) m= 2xy => x2 – y2 =4y -(1)
    and   12/(2x-y) - 12/(2x+y) =1 => 4x2 – y2 = 24y
          x2= ( 24y + y2)/4 -->(2)
         from 1 and 2 we have
         4y+ y2 =( 24y+y2)/4 
           y=8/3 mph
           y= 22/3 mph
 
8.There is a road beside a river. two friends started from 
a place A, moved to a temple situated at another place B 
and then returned to A again. one of them moves on a cycle 
at a speed of 12 kmph, while the other sails on a boat at a
speed of 10 kmph . if the river flows at the speedof 4 kmph,
which of the two friends will return to place A first ?

Sol: Clearly, The cyclist moves both ways at a speed of 12 kmph
so, average speed of the cyclist = 12 kmph 
the boat sailor moves downstream = (10+4) = 14 kmph
upstream =(10-4) = 6 kmph
So, average speed of the boat sailor =[ 2*14*6]/[14+6] kmph
=42/5 kmph =8.4 kmph
Since, the average speed of the cyclist is greater, he will 
return to A first.

9.A boat takes 19 hrs for travelling downstream from point A to
point B. and coming back to a point C midway between A and B.
if the velocity of the sream is 4 kmph. and the speed of the 
boat in still water is 14 kmph. what is the distence between 
A and B?

Sol:
speed downstream =14+4 =18 kmph
speed upstream = 14 -4 = 10 kmph
let the distence between A and B be x km. then,
x/18 + (x/2)/10 = 19
x/18 + x/20 =19
19x/180 =19 =>x = 180km
Hence, the distence between A and B bw 180 km